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Logic Proofs PowerPoint PPT Presentation


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Logic Proofs. Monday 9/29/08. Take out HW – “Drawing Conclusions”; Pick up handouts! Quiz handed back Wednesday Logic Proofs - Law of Detachment HW – 1. Essay Due Thursday! 2. Law of Detachment. Definitions:.

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Logic Proofs

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Logic Proofs


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Monday 9/29/08

  • Take out HW – “Drawing Conclusions”; Pick up handouts!

  • Quiz handed back Wednesday

  • Logic Proofs

  • - Law of Detachment

  • HW – 1. Essay Due Thursday!

  • 2. Law of Detachment


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Definitions:

Logically equivalent: when two statements always have the same truth value.

Premise: the statement that is given and excepted to be true.

Conclusion: the statement that has come from the premises.


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Law of Detachment(a law of inference)

  • The Law of Detachment states that when two given premises are true, one a conditional and the other the hypothesis of that conditional it then follows that the conclusion of the conditional is true.

  • If you are given a conditional and the hypothesis the conclusion is true.

  • p q and p therefore q

End for today


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Wednesday 10/1/08

  • Take out HW – “Law of Detachment”; Pick up handout!

  • Place #1 – 4 answers (truth tables) on board! HW pass

  • 2. Discuss Quiz

  • 3. Logic Proofs

  • - The Law of Contrapositive

  • - Formal Proof

  • - Modus Tollens

  • HW – 1. Essay Due Thursday!

  • 2. Law of Contrapositive / Modus Tollens


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Law of Contrapositive

  • The Law of Contrapositive states that when a conditional premise is true, it follows that the contrapositive of the premise is also true.

  • If the conditional is true the contrapositive is also true.

  • p  q then ~q  ~p


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Formal Proof: given premises that are true use laws of reasoning to reach a conclusion.

Two Column Proof:

Column 1: Statements

Column 2: Reasons

Each statement must have a reason and they are number in sequence.


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Example:

Given: If Joanna saves enough money, then she can buy a bike. Joanna can not buy a bike.

Prove: Joanna did not have enough money.

Let m = “ Joanna saves enough money”

Let b = “ Joanna can buy a bike”

Given: m  b, ~b

Prove: ~m


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Statement

m  b

~b

Reason

Given

Given

3. Law of Contrapositive (step 1)

3. ~b  ~m

4. ~m

4. Law of Detachment (step 2 & 3)

End for today


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Quiz Review

1. Rewrite the following as an equivalent DISJUNCTION!

a. 2x ≠ 4

a. (2x < 4) ∨ (2x > 4)

b. (b - 2 < 2) ∨ (b - 2 > 2)

b. b - 2 ≠ 2

  • Write the negation of each in simplest terms!

  • A. 10 + x < 5

  • B. I never fail quizzes.

  • C. Math is logical.

  • D. It is not true that I am a good baseball player.

A. 10 + x ≥ 5

B. I sometimes fail quizzes

C. Math is not Logical.

D. I am a good baseball player.


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Quiz Review

  • Write the following in symbolic form using the letters given. In each case the letter is the positive value

  • of the statement!

a. The absolute value of x is equal to 2 if and

only if x =2 or x = -2.

(p: absolute value of x is 2; q: x = 2; r: x = -2)

b. If I’m late, then I’ll get into trouble, and if I’m

not late, then I won’t get into trouble.. (l,t)

Answers placed on board!


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Thursday 10/2/08

  • Take out HW – “Law of Contrapositive”; Place essay in box!

  • Complete Pre-game Warm-up! See Quiz Review

  • 3. Logic Proofs

  • - Modus Tollens

  • - Chain Rule (Law of Syllogism)

  • HW – 1. Law of Contrapositive / Modus Tollens #21-30

  • 2. Complete Chain Rule Worksheet


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Law of Modus Tollens

Given: If the tickets are sold out (t), then we’ll wait for the next show. (w)

We do not wait for the next show.

Prove: The tickets were not sold out.

Write the problem in symbolic notation!


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Law of Modus Tollens

Given:t →w t → w

~w ~ w

Prove:~ t ~ t

or [(t → w) Λ ~ w] → ~ t

Set up a truth table to prove!


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Prove [(t → w) Λ ~ w] → ~ t


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Prove [(t → w) Λ ~ w] → ~ t

[(t → w) Λ ~ w] → ~ t is a Tautology therefore a valid argument!


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Law of Modus Tollens

  • The law of Modus Tollens states when 2 given premises are true, one a conditional and the other the negation of the conclusion of that conditional, it then follows that the negation of the hypothesis of the conditional is true.

  • Given a conditional and the negation of the conclusion then the negation of the hypothesis is true. (combining Law of Contrapositive and Law of Detachment)

  • p  q, ~q therefore ~p


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Invalid Arguments

All premises are true but they do not all lead to a valid argument.

Example:pq

q

no conclusion

pq

~p

no conclusion

End for Today


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Friday 10/3/08

  • Take out HW – “Law of Contrapositive, Chain Rule worksheet”

  • Complete Quiz Retake! See Quiz Reviewhandout

  • 3. Logic Proofs

  • - Discuss Valid arguments

  • - Chain Rule (Law of Syllogism); Applications of chain rule

  • - Law of Disjunctive Inference

  • HW – 1. Law of Disjunctive Inference worksheet

  • 2. Application of Chain Rule #2-40 (evens)


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Quiz Review

1. Rewrite the following as an equivalent DISJUNCTION!

-x + 5 ≠ -3

  • Write the negation of each in simplest terms!

  • A. x ≤ -5

  • B. Monkeys do not live in trees.

  • C. Math is never logical.


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Quiz Review

  • Write the following in symbolic form using the letters given. In each case the letter is the positive value

  • of the statement!

If I don’t go to practice and train hard, then I will not be prepared for the game next week and our team will not win (p,h,g,w)


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Chain Rule (Law of Syllogism)

[(p → q) Λ(q → r)] → (p → r)


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Chain Rule (Law of Syllogism)

[(p → q) Λ(q → r)] → (p → r)


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Chain Rule (Law of Syllogism)

[(p → q) Λ(q → r)] → (p → r)


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Valid Arguments

An argument is valid if the implication (P1 P2 P3 P4 …. Pn) C.

Λ

Λ

Λ

Λ

Λ

Premises

Premises

Is a valid argument?

[(p → q) Λ ~ q] → ~ p

Law of Modus Tollens


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Chain RuleLaw of Syllogism

  • Chain rule states that when 2 given premises are true conditional such that the consequent (conclusion) of the 1st is the antecedent (hypothesis) of the 2nd it follows that a conditional formed using the antecedent of the 1st and the consequent of the 2nd is true.

  • You can combine conditionals if the conclusion of one is the hypothesis of another.

  • p  q and q r then p  r


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Chain RuleLaw of Syllogism

p q

q  r

p  r

[(p → q) Λ(q → r)] → (p → r)


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Chain RuleExample

p : You study

q : You pass

r : You get a surprise

P1:

p  q

If you study, then you will pass.

P2:

q  r

If you pass, then you will get a surprise.


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Chain RuleExample

p : You study

q : You pass

r : You get a surprise

P1:

p  q

If you study, then you will pass.

P2:

q  r

If you pass, then you will get a surprise.

p  r

C:

If you study, then you will get a surprise.


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Law of Disjunction

  • The previous laws involved conditionals this one does not.

  • Law of Disjunctive Inference states when 2 given premises are true, on a disjunction and the other the negation of one of the disjuncts it then follows the other disjunct is true.

    p ν qorp ν q

    ~q ~p

 p

 q

End for Today


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Monday 10/6/08

  • Take out HW – “Law of Disjuctive Inference, Chain Rule worksheet & Chain Rule Applications” – Place proofs on the white board!

  • Another Proof - #10 Disjunctive Inference

  • 3. Logic Proofs

  • - Double Negation

  • - DeMorgan’s Law

  • 4. Begin HW; Pass back Quizzes

  • HW – 1. pg 86-87 (evens)

  • 2. Parent Signatures (3/2- by tomorrow!)


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#10 Hw Disjunctive Inference

Givens:

Don is first or Nancy is Second.

If Nancy is second, then Chris is third.

If Chris is third, then Pattie is fourth.

Pattie is not fouth.

Prove: Pattie is not fourth.


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Negations

  • Contrapositive,

    ~p q  ~q ~(~p)

    ~p q  ~q  p

  • Modus Tollens

    {(~p q) Λ ~q}  ~(~p)

    {(~p q) Λ ~q}  p


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Law of Double Negation

  • Law of double negation states ~(~p) and p are logically equivalent.

  • (you do not need to apply this law, continue as we have been)


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DeMorgan’s Laws

-Discovered by English mathematician

-Tells us how to negate a conjunction and disjunction

Complete the following truth table


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Since ~(m^s) and ~mV~s are logically equivalent for all cases, the negation of a conjunction is the disjunction.


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DeMorgan’s Law states

  • The negation of a conjunction of 2 statements is logically equivalent to the disjunction of the negation of each of the two statements.

  • The negation of a disjunction of 2 statements is logically equivalent to the conjunction of the negation of each of the 2 statements.

  • ~(p ^ q)  (~p V ~q)

  • ~(pV q )   (~p ^~q)

End for today


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Laws of Simplification

Mike likes to read and play basketball.

We can conclude-

Mike likes to read.

Mike likes to play basketball.


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Laws of Simplification

  • The law of simplification states that when a single conjunctive premise is true, it follows that each of the individual conjucts must be true.

  • p ^ q therefore p is true

  • p ^ q therefore q is true


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Law of Conjunction:

The law of conjunction states that when 2 given premises are true, it follows that the conjunction of these is true.

p

q

p ^ q


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Law of Conjunction:

AB is perpendicular to CD

AB is the bisector of CD

AB is the perpendicular bisector of CD.


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Law of Disjunctive Addition

  • Law of disjunctive addition states that when a single premise is true, it follows that any disjunction that has this premise as a disjunct is also true.

    p

    p V q


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Law of Disjunctive Addition

We solve an equation and see that x = 5.

We may also include that the statement

x = 5 or x > 5 is

True.

We may also include that the statement

x = 5 or Ian has 3 eyes is

True.


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Other Problems:

Given the statement r, which is a valid conclusion:

1. r k 2. r k 3. r k 4. r k

V

^


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Wednesday 10/8/08

  • Take out HW – “Law of Simplification,…” – Place proofs on the white board!

  • 2. Logic Proofs – Practice!!!

  • HW – Logic Proof Quiz Friday!


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